Solve for $x$ : $ 5|x - 6| - 8 = -4|x - 6| + 3 $
Solution: Add $ {4|x - 6|} $ to both sides: $ \begin{eqnarray} 5|x - 6| - 8 &=& -4|x - 6| + 3 \\ \\ { + 4|x - 6|} && { + 4|x - 6|} \\ \\ 9|x - 6| - 8 &=& 3 \end{eqnarray} $ Add ${8}$ to both sides: $ \begin{eqnarray} 9|x - 6| - 8 &=& 3 \\ \\ { + 8} &=& { + 8} \\ \\ 9|x - 6| &=& 11 \end{eqnarray} $ Divide both sides by ${9}$ $ \dfrac{9|x - 6|} {{9}} = \dfrac{11} {{9}} $ Simplify: $ |x - 6| = \dfrac{11}{9}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x - 6 = -\dfrac{11}{9} $ or $ x - 6 = \dfrac{11}{9} $ Solve for the solution where $x - 6$ is negative: $ x - 6 = -\dfrac{11}{9} $ Add ${6}$ to both sides: $ \begin{eqnarray} x - 6 &=& -\dfrac{11}{9} \\ \\ {+ 6} && {+ 6} \\ \\ x &=& -\dfrac{11}{9} + 6 \end{eqnarray} $ Change the ${ + 6}$ to an equivalent fraction with a denominator of $9$ $ x = - \dfrac{11}{9} {+ \dfrac{54}{9}} $ $ x = \dfrac{43}{9} $ Then calculate the solution where $x - 6$ is positive: $ x - 6 = \dfrac{11}{9} $ Add ${6}$ to both sides: $ \begin{eqnarray} x - 6 &=& \dfrac{11}{9} \\ \\ {+ 6} && {+ 6} \\ \\ x &=& \dfrac{11}{9} + 6 \end{eqnarray} $ Change the ${ + 6}$ to an equivalent fraction with a denominator of $9$ $ x = \dfrac{11}{9} {+ \dfrac{54}{9}} $ $ x = \dfrac{65}{9} $ Thus, the correct answer is $x = \dfrac{43}{9} $ or $x = \dfrac{65}{9} $.